On the Role of the Deterministic and Circumferential Stresses in Throughflow Calculations

This paper presents a throughflow analysis tool developed in the context of the average-passage flow model elaborated by Adamczyk. The Adamczyk’s flow model describes the 3-D timeaveraged flow field within a blade row passage. The set of equations that governs this flow field is obtained by performing a Reynolds averaging, a time averaging and a passage-to-passage averaging on the Navier-Stokes equations. The throughflow level of approximation is obtained by performing an additional circumferential averaging on the 3-D average-passage flow. The resulting set of equations is similar to the 2-D axisymmetric Navier-Stokes equations but additional terms resulting from the averages show up : blade forces, blade blockage factor, Reynolds stresses, deterministic stresses, passage-to-passage stresses and circumferential stresses. This set of equations represents the ultimate throughflow model provided that all stresses and blade forces can be modeled. The relative importance of these additional terms is studied in the present contribution. The stresses and the blade forces are determined from 3-D steady and unsteady databases (a low speed compressor stage and a transonic turbine stage) and incorporated in a throughflow model based on the axisymmetric Navier-Stokes equations. A good agreement between the throughflow solution and the averaged 3-D results is obtained. These results are also compared to those obtained with a more “classical” throughflow approach based on a Navier-Stokes formulation for the endwall losses, correlations for profile losses and a simple radial mixing model assuming turbulent diffusion. NOMENCLATURE b blockage factor fb inviscid blade force fv viscous blade force h, H enthalpy, total enthalpy k kinetic energy of the fluctuations p, p0 pressure, total pressure q heat flux s entropy (relative to inlet conditions) T 0 total temperature V absolute velocity x, r, θ axial,radial and circumferential coordinates α absolute flow angle ρ density τ viscous stress Operators averaging operator ̃ Favre averaging operator Subscripts x, r, θ axial, radial and circumferential components R relative to a rotor S relative to a stator Superscripts ′ unsteady non-deterministic fluctuation ′′ unsteady deterministic fluctuation ′′′ aperiodic fluctuation ′′′′ circumferential fluctuation 1 Copyright c © 2008 by ASME INTRODUCTION The throughflow level of approximation still remains an important tool for designing turbomachines [10] even though 3-D calculations are used more and more early in the design process. Throughflow codes are mainly used at the preliminary design stage for specifying the target aerodynamic performances to be achieved by the blading. They can be used either in the design mode, where the angular momentum and/or the total conditions are prescribed and the flow angles are sought, either in the analysis mode, where a known machine geometry is analyzed for its performance. The throughflow models are also used to exploit experimental results or to couple single blade row calculations in order to compute the flow field inside a multistage machine [4]. Unfortunately, these models heavily rely on empirical inputs, such as profile losses correlations or end-wall loss models. They can accurately predict the flow field inside a turbomachine provided that the design parameters are not too far from those of the reference machines that were used to calibrate the throughflow model. This approach has shown to be efficient but lacks of generality. The most widespread throughflow method is certainly the streamline curvature method (SLC). In 1980, Spuur has proposed another approach based on the Euler equations. This approach has only started to retain attention in the 1990. Recent works using this approach can be found in [5] and [22]. The methods based on the Euler equations present some interesting features and eliminate some of the drawbacks of the streamline curvature approach, such as a shock capturing property or a natural unsteady capibility with the generally adopted time-marching technique. However the treatment of the annulus endwalls is probably the major concern for the throughflow models based on the SLC method as well as the ones based on the Euler equations. A common practice is to introduce an aerodynamic blockage equivalent to the displacement thickness of the endwall boundary layers which corrects the mass flow in order to obtain the correct level of velocity in the core flow. The blockage factor is a very sensitive quantity relying on empiricism for which a misprediction can lead to a mismatch of the stages. Another solution is to perform a separate boundary layer calculation. However, it is recognized that the use of the boundary layer theory for computing the endwall flows inside a turbomachine (especially in a compressor) is inappropriate [9], [14]. Recently another solution has been proposed by the authors with a throughflow model directly based on the Navier-Stokes equations. It is able to resolve the viscous flow on the annulus endwalls and the corresponding aerodynamic blockage. It can also capture 2-D recirculations. The θ-averaged Navier-Stokes equations are solved by a finite volume technique. By including more physics in the model, less empiricism is needed and a more general method can be devised. Details of this model as well as the numerical techniques used can be found in [17] and [18]. In the present contribution, the authors propose another step toward less empiricism in throughflow calculations with a highorder throughflow method. This model is based on the Adamczyk cascade of averaging procedure [1]. Adamczyk addresses the 3-D unsteady and turbulent flow field via several averaging operations. The first one is the well known Reynolds averaging which eliminates the effects of the turbulence, leaving a deterministic unsteady flow. The second one is a time averaging removing the remaining effects of the unsteadiness due to the movement of the rotor blades with respect to the stator ones. Finally an aperiodic averaging eliminates the aperiodicity of the flow due to the blade indexing. The resulting flow field is steady and periodic but it incorporates the mean effects of the turbulence, the unsteadiness and the aperiodicity. The equations associated to this flow show different unknown terms bringing the aforementioned effects. These terms are the Reynolds stresses, the deterministic stresses, the passageto-passage stresses and the blade forces. These equations, which have been rigorously obtained, are the so-called average-passage equations and describe the steady flow field inside a blade row embedded in a multistage environment. In this contribution a subsequent step is performed by circumferentially averaging the average-passage equations in order to obtain an axisymmetric representation of the flow. These equations are rigorously obtained and contain the effects of the non-axisymmetry of the flow through circumferential stresses and blade forces. This set of equations represents the ultimate throughflow model. The sole assumptions are those prevailing to the establishment of the Navier-Stokes equations. However these throughflow equations present several unknown terms which have to be modelled or closed. This tremendous task is far beyond the scope of this paper. In the present contribution, the relative importance of the different terms of such a throughflow calculation is studied as well as the benefit brought by this model compared to more classical throughflow models. These analysis are performed with the help of 2-D and 3-D test-cases. The development of a boundary layer over a flat plate and the spreading of a wake allow to highlight the main properties of the averaged equations and to perform a first attempt to evaluate the relative importance of the different unknowns brought by the averaging process. The high-order throughflow model is also applied to two turbomachine test-cases, a low speed compressor stage and a transonic turbine stage. These test-cases reside in 3-D steady and unsteady numerical simulations from which the different terms needed for the closure of the throughflow equations are extracted. They allow to further study the high-order throughflow model and to show the improvements compared to a classical throughflow. 2 Copyright c © 2008 by ASME CIRCUMFERENTIAL-AVERAGED EQUATIONS The average-passage set of equations of Adamczyk is obtained by successively averaging the unsteady Navier-Stokes equations on an ensemble of realizations in the sense of Reynolds on time and on the passages of a given blade row. This triple averaging procedure brings the mean effects of the turbulence, the unsteadiness and the aperiodicity on the steady flow field inside a blade row embedded in a multistage configuration. These effects appear as additional terms in the conservation equations, namely Reynolds stresses, deterministic stresses, passage-to-passage stresses, blade blockage factors and blade forces. For example, for a given stator j embedded in a multistage machine, the resulting axial momentum equation is written as follows: